Functions: Fractional functions
Power functions with negative exponents
Power function with negative exponent
A power function with a negative integer exponent has the form \[f(x)=\blue{a}x^{-\orange{n}}\]
in which #\orange{n}# is a postive integer.
We can also write this function as \[f(x)=\frac{\blue{a}}{x^{\orange{n}}}\]
The graph of a power function with a negative integer exponent, moves through the point #\rv{1,\blue{a}}#, has a vertical asymptote at #x=0# and horizontal asymptote in the line #y=0#.
If #\orange{n}# is even, the function is symmetrical across the #y#-axis. If #\orange{n}# is odd, the function has the point #\rv{0,0}# as the point of symmetry.
GeoGebra Negative powerfunction
Take a look at the graph of a power function with negative exponent, which is a function of the form #f(x)=\frac{a}{x^n}#.
What do we know about the values of #n# and #a#?
What do we know about the values of #n# and #a#?
The value of #n# is: odd
The value of #a# is: negative
The graph is symmetrical across the point #\rv{0,0}#, hence, the value of #n# is odd.
The #y#-value is negative if the value of #x# is positive, hence, the value of #a# is negative.
The value of #a# is: negative
The graph is symmetrical across the point #\rv{0,0}#, hence, the value of #n# is odd.
The #y#-value is negative if the value of #x# is positive, hence, the value of #a# is negative.
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